Extremal Solutions and Comparison Principle for Nonlinear Integro-Differential Equations in a Banach Space

<正> In this paper, we consider an initial value problem for nonlinear integro-differentialequations in a Banach space. First, we give a comparison result between the under and overfunctions and some comparison principles. Then, using these results and the Kuratowski measureof noncompactness, we establish the existence theorem of extremal solutions between the underand over functions, and prove that there exists a unique solution between the lower and uppersolutions under an additional Lipschitz's condition.